In computing, half precision (sometimes called FP16) is a binary floating-point computer number format that occupies 16 bits (two bytes in modern computers) in computer memory. It is intended for storage of floating-point values in applications where higher precision is not essential, in particular image processing and neural networks.
Almost all modern uses follow the IEEE 754-2008 standard, where the 16-bit base-2 format is referred to as binary16, and the exponent uses 5 bits. This can express values in the range ±65,504, with the minimum value above 1 being 1 + 1/1024.
Depending on the computer, half-precision can be over an order of magnitude faster than double precision, e.g. 550 PFLOPS for half-precision vs 37 PFLOPS for double precision on one cloud provider.
Several earlier 16-bit floating point formats have existed including that of Hitachi's HD61810 DSPof 1982, Scott's WIF and the 3dfx Voodoo Graphics processor.
ILM was searching for an image format that could handle a wide dynamic range, but without the hard drive and memory cost of single or double precision floating point.The hardware-accelerated programmable shading group led by John Airey at SGI (Silicon Graphics) invented the s10e5 data type in 1997 as part of the 'bali' design effort. This is described in a SIGGRAPH 2000 paper (see section 4.3) and further documented in US patent 7518615. It was popularized by its use in the open-source OpenEXR image format.
Nvidia and Microsoft defined the half datatype in the Cg language, released in early 2002, and implemented it in silicon in the GeForce FX, released in late 2002. [ citation needed ]Since then support for 16-bit floating point math in graphics cards has become very common.
The F16C extension in 2012 allows x86 processors to convert half-precision floats to and from single-precision floats with a machine instruction.
The IEEE 754 standardspecifies a binary16 as having the following format:
The format is laid out as follows:
The format is assumed to have an implicit lead bit with value 1 unless the exponent field is stored with all zeros. Thus only 10 bits of the significand appear in the memory format but the total precision is 11 bits. In IEEE 754 parlance, there are 10 bits of significand, but there are 11 bits of significand precision (log10(211) ≈ 3.311 decimal digits, or 4 digits ± slightly less than 5 units in the last place).
The half-precision binary floating-point exponent is encoded using an offset-binary representation, with the zero offset being 15; also known as exponent bias in the IEEE 754 standard.
Thus, as defined by the offset binary representation, in order to get the true exponent the offset of 15 has to be subtracted from the stored exponent.
The stored exponents 000002 and 111112 are interpreted specially.
|Exponent||Significand = zero||Significand ≠ zero||Equation|
|000002||zero, −0||subnormal numbers||(−1)signbit × 2−14 × 0.significantbits2|
|000012, ..., 111102||normalized value||(−1)signbit × 2exponent−15 × 1.significantbits2|
|111112||±infinity||NaN (quiet, signalling)|
The minimum strictly positive (subnormal) value is 2−24 ≈ 5.96 × 10−8. The minimum positive normal value is 2−14 ≈ 6.10 × 10−5. The maximum representable value is (2−2−10) × 215 = 65504.
These examples are given in bit representation of the floating-point value. This includes the sign bit, (biased) exponent, and significand.
|0 00000 0000000000||0000||0|
|0 00000 0000000001||0001||2−14 × (0 + 1/1024 ) ≈ 0.000000059604645||smallest positive subnormal number|
|0 00000 1111111111||03ff||2−14 × (0 + 1023/1024 ) ≈ 0.000060975552||largest subnormal number|
|0 00001 0000000000||0400||2−14 × (1 + 0/1024 ) ≈ 0.00006103515625||smallest positive normal number|
|0 01101 0101010101||3555||2−2 × (1 + 341/1024 ) ≈ 0.33325195||nearest value to 1/3|
|0 01110 1111111111||3bff||2−1 × (1 + 1023/1024 ) ≈ 0.99951172||largest number less than one|
|0 01111 0000000000||3c00||20 × (1 + 0/1024 ) = 1||one|
|0 01111 0000000001||3c01||20 × (1 + 1/1024 ) ≈ 1.00097656||smallest number larger than one|
|0 11110 1111111111||7bff||215 × (1 + 1023/1024 ) = 65504||largest normal number|
|0 11111 0000000000||7c00||∞||infinity|
|1 00000 0000000000||8000||−0|
|1 10000 0000000000||c000||-2|
|1 11111 0000000000||fc00||−∞||negative infinity|
By default, 1/3 rounds down like for double precision, because of the odd number of bits in the significand. The bits beyond the rounding point are 0101... which is less than 1/2 of a unit in the last place.
65519 is the largest number that will round to a finite number (65504), 65520 and larger will round to infinity. This is for round-to-even, other rounding strategies will change this cutoff.
ARM processors support (via a floating point control register bit) an "alternative half-precision" format, which does away with the special case for an exponent value of 31 (111112).It is almost identical to the IEEE format, but there is no encoding for infinity or NaNs; instead, an exponent of 31 encodes normalized numbers in the range 65536 to 131008.
This format is used in several computer graphics environments to store pixels, including MATLAB, OpenEXR, JPEG XR, GIMP, OpenGL, Cg, Direct3D, and D3DX. The advantage over 8-bit or 16-bit integers is that the increased dynamic range allows for more detail to be preserved in highlights and shadows for images, and the linear representation of intensity making calculations easier. The advantage over 32-bit single-precision floating point is that it requires half the storage and bandwidth (at the expense of precision and range).
Hardware and software for machine learning or neural networks tend to use half precision: such applications usually do a large amount of calculation, but don't require a high level of precision.
If the hardware has instructions to compute half-precision math, it is often faster than single or double precision. If the systems has SIMD instructions that can handle multiple floating-point numbers within one instruction, half precision can be twice as fast by operating on twice as many numbers simultaneously.However, if there is no hardware support, math must be done by emulation, or by conversion to single or double precision and then back, and is therefore slower.
Several versions of the ARM architecture have support for half precision.
Support for half precision in the x86 instruction set is specified in the AVX-512_FP16 instruction set extension to be implemented in the future Intel Sapphire Rapids processor.
In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. For this reason, floating-point computation is often used in systems with very small and very large real numbers that require fast processing times. In general, a floating-point number is represented approximately with a fixed number of significant digits and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form:
IEEE 754-1985 was an industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point libraries, and in hardware, in the instructions of many CPUs and FPUs. The first integrated circuit to implement the draft of what was to become IEEE 754-1985 was the Intel 8087.
A computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable computers and calculators. Numerical values are stored as groupings of bits, such as bytes and words. The encoding between numerical values and bit patterns is chosen for convenience of the operation of the computer; the encoding used by the computer's instruction set generally requires conversion for external use, such as for printing and display. Different types of processors may have different internal representations of numerical values and different conventions are used for integer and real numbers. Most calculations are carried out with number formats that fit into a processor register, but some software systems allow representation of arbitrarily large numbers using multiple words of memory.
Double-precision floating-point format is a computer number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.
In computer science, subnormal numbers are the subset of denormalized numbers that fill the underflow gap around zero in floating-point arithmetic. Any non-zero number with magnitude smaller than the smallest normal number is subnormal.
The IEEE Standard for Floating-Point Arithmetic is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard.
The significand is part of a number in scientific notation or in floating-point representation, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.
Hexadecimal floating point is a format for encoding floating-point numbers first introduced on the IBM System/360 computers, and supported on subsequent machines based on that architecture, as well as machines which were intended to be application-compatible with System/360.
In IEEE 754 floating-point numbers, the exponent is biased in the engineering sense of the word – the value stored is offset from the actual value by the exponent bias, also called a biased exponent. Biasing is done because exponents have to be signed values in order to be able to represent both tiny and huge values, but two's complement, the usual representation for signed values, would make comparison harder.
In C and related programming languages,
long double refers to a floating-point data type that is often more precise than double precision though the language standard only requires it to be at least as precise as
double. As with C's other floating-point types, it may not necessarily map to an IEEE format.
In computing, minifloats are floating-point values represented with very few bits. Predictably, they are not well suited for general-purpose numerical calculations. They are used for special purposes, most often in computer graphics, where iterations are small and precision has aesthetic effects. Machine learning also uses similar formats like bfloat16. Additionally, they are frequently encountered as a pedagogical tool in computer-science courses to demonstrate the properties and structures of floating-point arithmetic and IEEE 754 numbers.
Extended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. In contrast to extended precision, arbitrary-precision arithmetic refers to implementations of much larger numeric types using special software.
Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions and binary (base-2) fractions.
IEEE 754-2008 was published in August 2008 and is a significant revision to, and replaces, the IEEE 754-1985 floating-point standard, while in 2019 it was updated with a minor revision IEEE 754-2019. The 2008 revision extended the previous standard where it was necessary, added decimal arithmetic and formats, tightened up certain areas of the original standard which were left undefined, and merged in IEEE 854.
In computing, quadruple precision is a binary floating point–based computer number format that occupies 16 bytes with precision at least twice the 53-bit double precision.
Single-precision floating-point format is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.
In computing, decimal64 is a decimal floating-point computer numbering format that occupies 8 bytes in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations.
In computing, Microsoft Binary Format (MBF) is a format for floating-point numbers which was used in Microsoft's BASIC language products, including MBASIC, GW-BASIC and QuickBASIC prior to version 4.00.
In computing, octuple precision is a binary floating-point-based computer number format that occupies 32 bytes in computer memory. This 256-bit octuple precision is for applications requiring results in higher than quadruple precision. This format is rarely used and very few environments support it.
The bfloat16 floating-point format is a computer number format occupying 16 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. This format is a truncated (16-bit) version of the 32-bit IEEE 754 single-precision floating-point format (binary32) with the intent of accelerating machine learning and near-sensor computing. It preserves the approximate dynamic range of 32-bit floating-point numbers by retaining 8 exponent bits, but supports only an 8-bit precision rather than the 24-bit significand of the binary32 format. More so than single-precision 32-bit floating-point numbers, bfloat16 numbers are unsuitable for integer calculations, but this is not their intended use. Bfloat16 is used to reduce the storage requirements and increase the calculation speed of machine learning algorithms.
Nvidia recently introduced native half precision floating point support (FP16) into their Pascal GPUs. This was mainly motivated by the possibility that this will speed up data intensive and error tolerant applications in GPUs.
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